Elsevier

Automatica

Volume 44, Issue 8, August 2008, Pages 2171-2178
Automatica

Brief paper
Decentralized control design of interconnected chains of integrators: A case study

https://doi.org/10.1016/j.automatica.2007.12.011Get rights and content

Abstract

We develop a constructive decentralized control design procedure for a class of systems that may be loosely described as chained integrators which are dynamically coupled. The design method is inspired by nested saturation control ideas and formulated by applying the singular perturbation theory. We demonstrate that the proposed design provides a Lyapunov function for an associated closed loop system from which semi-global stability may be deduced. Using the proposed idea, we design a semi-globally stabilizing control law for a four degree of freedom spherical inverted pendulum.

Introduction

Nested saturation control, useful in a nonlinear system with a forwarding structure, is associated with chains of integrators (Arcak et al., 2001, Grognard et al., 1999, Kaliora and Astofi, 2004, Kaliora and Astofi, 2005, Marconi and Isidori, 2000, Teel, 1996). This often leads to a slow closed loop response. Indeed, its transients exhibit a time-scale separation between various “nested” controllers, which is not inherent in the nonlinear system itself. It appears that some structure, without necessarily emulating the conservativeness of these nested saturating controllers, can be achieved using linear control ideas combined with singular perturbation tools that exploit natural time scales.

Here, we show how linear control ideas with time scaling recover many properties inherent in the nested saturation design. Our design is constructive and comes with a Lyapunov function for formally stating stability and robustness. In this regard, our paper extends the work by Grognard, Sepulchre, Bastin, and Praly (1998) and Mazenc (1997) studying a single input single chain of integrators. In our approach, time scales are selected “per block of states” and not for each state component on succession. A spherical inverted pendulum is a beam attached to a horizontal plane via a universal joint that is free to move in the plane under the influence of a planar force (see Fig. 1). The pendulum in the upper space is assumed. Its modelling was given in Liu (2006); its non-local stabilization and output tracking were first explicitly solved in Liu et al., 2008a, Liu et al., 2008b respectively. We design a stabilizing controller using the proposed idea for this pendulum that achieves an arbitrarily large domain of attraction in the upper space by tuning a scaling parameter. Beside recovering many features in the nested saturating controller (Liu et al., 2008a), it exploits natural time scaling and hence is less conservative. The case study is a representative from a large class of mechanical systems that can be viewed as dynamically coupled chains of integrators. It is for this family that we propose a decentralized control strategy.

Section snippets

Notation

Let a vector v(v1T,v2T,,vnT)TRn1×Rn2××RnN. For a vector vjRnj, vi,j, i=1,,nj, denotes ith element of vj. With a polynomial sn+ansn1++a2s+a1, we associate a companion matrix: A=(01001a1a2an).s() and c() denote sin() and cos() respectively. The methodology used here is based on standard singular perturbation tools (see Kokotović, Khalil, and O’Reilly (1986) for details) to an autonomous singularly perturbed system ẋ=f(x,z,ε),εż=g(x,z,ε),for ε>0, where xDxRn, zDzRm,

Problem statement

We consider a sequence of N interconnected chains of integrators where each subsystem j{1,2,,N} consists of two blocks as follows Σx,j:ẋ1,j=x2,j+φ1,j(y),,ẋnj,j=y1,j+φnj,j(y),Σy,j:ẏ1,j=y2,j,,ẏmj1,j=ymj,j,ẏmj,j=uj. Let state vectors be x(x1,,xN)=(x1,1,,xn1,1,,x1,N,,xnN,N)Rn1××RnN and y(y1,,yN)=(y1,1,,ym1,1,,y1,N,,ymN,N)Rm1××RmN and an input vector be u(u1,,uN)RN. φi,j(), i{1,2,,nj} and j{1,2,,N}, are zero at y=0, analytic and higher order terms with respect to y in

The model

Refer to Fig. 1. Consider a spherical inverted pendulum denoted by a set of generalized coordinates q(x,y,δ,ϵ) in a configuration space UR×R×(π/2,π/2)×(π/2,π/2). F=(FxFy)T is a planar control signal applied to a pivot attached to the bottom of the pendulum. Unknown exogenous inputs are collected by vfR4. We review the model in Liu et al. (2008a) as follows, D(q)q̈+C(q,q̇)q̇+G(q)=Q, where D(q), C(q,q̇), G(q) and Q are given in Appendix.

Decentralized control design

The nominal dynamics of (17) with the exogenous input

Summary

A decentralized linear control scheme is proposed for certain systems possessing interconnected chains of integrators and is applied to a spherical inverted pendulum. The corresponding closed loop systems yield some arbitrarily large domains of attraction by adjusting a design parameter, which is guaranteed by the associated Lyapunov functions.

Dr. Guangyu Liu received his Ph.D. degree in dynamics and control from The University of Melbourne, Australia in 2006. He was a mechanical engineer during 1997–2001 at Dong-feng motor corporation (NISSAN), China and a researcher at a micro-fabrication lab, Australia in 2002. He is presently a researcher at NICTA, Australia. His research interests are dynamics, decision and control and various applications in both engineering and life science.

References (16)

There are more references available in the full text version of this article.

Cited by (18)

  • Tracking problems of a spherical inverted pendulum via neural network enhanced design

    2013, Neurocomputing
    Citation Excerpt :

    So far, various control problems including stabilization and output regulation of this system have attracted some attentions [1,4,11–15,17,18].

  • Stabilization and tracking control of X-Z inverted pendulum with sliding-mode control

    2012, ISA Transactions
    Citation Excerpt :

    Until recently, there are a lot of literatures on the swing up, stabilization, and tracking control of the traditional inverted pendulum. Beside the wide research on the traditional inverted pendulum, some researchers concentrate their efforts on the other types of inverted pendulums, such like spherical inverted pendulum (which can also be named as X–Y inverted pendulum, planar inverted pendulum or dual-axis inverted pendulum) [4,5,11], X–Z inverted pendulum [6–8] and inverted 3-D pendulum [9]. The X–Z inverted pendulum can move in the vertical plane with horizontal and vertical forces which is first proposed by Maravall [6,7].

  • Approximate output regulation of spherical inverted pendulum by neural network control

    2012, Neurocomputing
    Citation Excerpt :

    The stabilization problem of the spherical inverted pendulum was considered in several papers [1–3,7,16,18,22,23].

  • A note on the control of a spherical inverted pendulum

    2007, IFAC Proceedings Volumes (IFAC-PapersOnline)
  • Extension Principle and Controller Design for Nonlinear Systems

    2023, 2023 European Control Conference, ECC 2023
View all citing articles on Scopus

Dr. Guangyu Liu received his Ph.D. degree in dynamics and control from The University of Melbourne, Australia in 2006. He was a mechanical engineer during 1997–2001 at Dong-feng motor corporation (NISSAN), China and a researcher at a micro-fabrication lab, Australia in 2002. He is presently a researcher at NICTA, Australia. His research interests are dynamics, decision and control and various applications in both engineering and life science.

Dr. Iven Mareels obtained his Ph.D. degree in Systems Engineering from Australian National University, Australia in 1987. He is a Professor and Dean of Faculty of Engineering, The University of Melbourne. Prof. Mareels is a co-editor in chief for Systems & Control Letters, a Fellow of ATSE (Australia), IEEE (USA) and IEAust, a member of SIAM, Vice-Chair of the Asian Control Professors Association, Deputy Chair of the National Committee for Automation, Control and Instrumentation, Chair of the Australian Research Council’s panel of experts for Mathematics, Information and Communication sciences, a member of the Board of Governors of the Control Systems Society IEEE and so on. He is registered as a professional engineer. His research interests are in adaptive and learning systems, nonlinear control and modeling.

Dr. Dragan Nešić is a Professor at The University of Melbourne, Australia. He received his Ph.D. degree in Systems Engineering from Australian National University, Australia in 1997. In 1997–1999, he held postdoctoral positions at University of California, Santa Barbara and two other institutions. Since 1999 he has been with The University of Melbourne. His research interests include networked control systems, discrete-time, sampled-data and continuous-time nonlinear control systems, input-to-state stability, extremum seeking control and so on. Prof. Nešić is a Fellow of IEEE, a Fellow of IEAust, a recipient of a Humboldt Research Fellowship (2003) and an Australian Professorial Fellow (2004–2009). He is an Associate Editor for the journals: Automatica, IEEE Transactions on Automatic Control, Systems and Control Letters and European Journal of Control.

This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Zongli Lin under the direction of Editor Hassan Khalil.

View full text